The matrix group $SO(n)$ can be treated as a submanifold of $\mathbf R^{n^2}$. Does it lie in any $(n^2-1)$-dimensional subspace of $\mathbf R^{n^2}$?
For $n=2$ the answer is yes because $SO(2)$ lies in the span of the identity matrix and $\begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}$. How about for $n>2$? Thanks.
It turns out that the linear hull of $SO(n)$ for $n>2$ is the whole $M_{n^2}(\mathbb R)$. Thus, the answer is no, there does not exist an $n^2-1$-dimensional subspace that contains $SO(n)$ for $n>2$.
Several proofs with further discussions may be found in this mathoverflow thread.